Finite Group Databasebeta

← #6624968

Subgroups — #6624968

Order 512.

1789 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#32960C212yesmaximal, complemented
5#29112C212yesmaximal, complemented
6#28115C212yesmaximal
7#26918C214yesmaximal, complemented
8#26774C214yesmaximal, complemented
9#26465C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic, complemented
11#25319C212yesmaximal, complemented
12#24209C211yesmaximal, characteristic
13#23221C212yesmaximal, complemented
14#23002C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16C4×#481C211yesmaximal, characteristic
17#21454C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#487C2212yes—
21C2×#481C2211yescharacteristic
22C2×#479C2214yes—
23C2×#461C2212yes—
24C2×#448C2212yes—
25C2×#440C2214yes—
26#2857C2211yescharacteristic
27#2851C2212yes—
28#2589—22no—
29#2563C2211yescharacteristic
30#2552C2211yescharacteristic
31#2509C2212yes—
32#2497—21no—
33#2487—21no—
34#2454C2212yes—
35#2450—22no—
36#2369C2211yescharacteristic
37#2365—22no—
38#2365—21no—
39#2365—21no—
40#2365C2214yes—
41#2365C2212yes—
42#2365C2212yes—
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2211yescharacteristic
45#2345—21no—
46C4×#121C2211yescharacteristic
47C4×#121C2212yes—
48C2×#391C2212yes—
49C2×#389—21no—
50C2×#387—22no—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2074C2214yes—
55#2066C2214yes—
56#2055C2214yes—
57#2043C2214yes—
58#2043C2212yes—
59#2026C2214yes—
60#2013C2212yes—
61#1987C2212yes—
62#1981C2212yes—
63#1956C2212yes—
64#1947C2212yes—
65#1939C2212yes—
66#1935C2214yes—
67#1935C2214yes—
68#1927C2214yes—
69#1926C2212yes—
70#1921C2214yes—
71#1919C2212yes—
72#1917C2214yes—
73#1880C2214yes—
74#1877C2214yes—
75#1874C2211yescharacteristic
76#1865C2214yes—
77#1863C2214yes—
78#1863C2214yes—
79#1853C2211yescharacteristic
80#1852C2212yes—
81#1824C2211yescharacteristic
82#1824C2212yes—
83#1805C2214yes—
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1787C2212yes—
87#1787C2212yes—
88#1787C2212yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×SD16C2314yes—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125C2312yes—
104C2×C22⋊Q8—21no—
105C2×#121C2311yescharacteristic
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—48no—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2314yes—
111C2×C4×D4—21no—
112C2×C4×D4—22no—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C42—22no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#487—28no—
128#481—24no—
129#479—48no—
130#461—28no—
131#448—28no—
132#440—48no—
133C2×#107C2314yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#400—22no—
153#400—22no—
154#399—22no—
155#398—22no—
156#396—22no—
157#393—22no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#391C2314yes—
164#390C2312yes—
165#390C2311yescharacteristic
166#389—22no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—24no—
171#389—21no—
172#389—21no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2311yescharacteristic
178#388—21no—
179#388—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#386—21no—
188#386—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#384—21no—
194#384C2311yescharacteristic
195#382C2312yes—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—22no—
201#380C2311yescharacteristic
202#380C2312yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—24no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C2×#95—22no—
222C2×#95—21no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—44no—
242C22×Q8C2414yes—
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2411yescharacteristic
255C2×SD16—28no—
256C2×SD16—28no—
257C2×SD16—44no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8#4118yes—
267#125—44no—
268C22⋊Q8—82no—
269#121—22no—
270#121#4114yes—
271C4×Q8—28no—
272C4×Q8—28no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—48no—
276C4×Q8—48no—
277C4×D4—24no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—84no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C42—24no—
331C2×C42—21no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—44no—
338C2×C42—48no—
339C2×C42—44no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—28no—
349#106—24no—
350#106—24no—
351C4⋊C8—24no—
352C4⋊C8—44no—
353#103—28no—
354#103—28no—
355#103—28no—
356#103—28no—
357#103—44no—
358#103—44no—
359#103—44no—
360#102—24no—
361#102—24no—
362C22⋊C8—24no—
363C22⋊C8—22no—
364C22⋊C8—22no—
365#97—44no—
366C4×C8—24no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—21no—
389C24C22×D411yescharacteristic
390C2×Q8—28no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—416no—
397C2×Q8—416no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×Q8C2×#4118yes—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C2×#4112yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4#13714yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C42−1+412yes—
459C22×C4C2511yescharacteristic
460SD16—416no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—42no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4C2×#4112yes—
503C4⋊C4H(5,𝔽2)14yes—
504C22⋊C4—24no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—81no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—24no—
517C42—24no—
518C42—42no—
519C42—42no—
520C42—48no—
521C42—48no—
522C42—44no—
523C42—42no—
524C42—44no—
525C42—82no—
526C42C2×#4114yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23#54812yes—
539C23C2×#13712yes—
540C23#57712yes—
541C23C23×D411yescharacteristic
542Q8—28no—
543Q8—28no—
544Q8—416no—
545Q8—44no—
546Q8—44no—
547Q8—162no—
548D4—24no—
549D4—48no—
550D4—42no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—416no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—164no—
607C2×C4#52512yes—
608C2×C4#52512yes—
609C2×C4#54812yes—
610C2×C4C2×#13712yes—
611C2×C4C2×#13712yes—
612C2×C4#57512yes—
613C2×C4C22×#4111yescharacteristic
614C2×C4C22×#4112yes—
615C2×C4C2×H(5,𝔽2)12yes—
616C2×C4C2×2−1+411yescharacteristic
617C8—42no—
618C8—44no—
619C8—42no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22#337214yes—
634C22C2×#52511yescharacteristic
635C22C2×#54811yescharacteristic
636C22#348411yescharacteristic
637C22C22×#13711yescharacteristic
638C22C2×#57511yescharacteristic
639C22C2×#57711yescharacteristic
640C22#358611yescharacteristic
641C4—22no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—41no—
651C4—48no—
652C4—81no—
653C4—82no—
654C4—81no—
655C4—82no—
656C4—82no—
657C4#311714yes—
658C4#348112yes—
659C4#352012yes—
660C2—24no—
661C2—82no—
662C2#3652811yescharacteristic
663C2C2×#311711yescharacteristic
664C2#6058211yescharacteristic
665C2C2×#337211yescharacteristic
666C2#6132311yescharacteristic
667C2#6158411yescharacteristic
668C2#6160411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2365C2semidirect
3. #33149C2semidirect
4. #32960C2semidirect
5. #29112C2semidirect
7. #26918C2semidirect
8. #26774C2semidirect
9. #26465C2semidirect
10. #25856C2semidirect
11. #25319C2semidirect
13. #23221C2semidirect
14. #23002C2semidirect
15. #22629C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#365512
C2×#23652
#331492
#329602
#291122
#281152
#269182
#267742
#264652
#258562
#253192
#242092
#232212
#230022
#226292
C4×#4812
#214542